Hilbert's third problem asks: do there exist two polyhedra
with the same volume which are not scissors congruent? In other
words, if P and Q are polyhedra with the same volume, is it always
possible to write P = ∪_{i=1}^{n} P_{i} and Q =∪_{i=1}^{n}
Q_{i} such that the P's and Q's intersect only on the boundaries
and such that P_{i} ≅ Q_{i}? In 1901 Dehn answered this question in
the negative by constructing a second scissors congruence invariant
now called the "Dehn invariant," and showing that a cube and a regular
tetrahedron never have equal Dehn invariants, regardless of their
volumes. We can then restate Hilbert's third problem: do the volume
and Dehn invariant separate the scissors congruence classes? In 1965
Sydler showed that the answer is yes; in 1968 Jessen showed that this
result extends to dimension 4, and in 1982 Dupont and Sah constructed
analogs of such results in spherical and hyperbolic geometries.
However, the problem remains open past dimension 4. By iterating Dehn
invariants Goncharov constructed a chain complex, and conjectured that
the homology of this chain complex is related to certain graded
portions of the algebraic K-theory of the complex numbers, with the
volume appearing as a regulator. In joint work with Jonathan
Campbell, we have constructed a new analysis of this chain complex
which illuminates the connection between the Dehn complex and
algebraic K-theory, and which opens new routes for extending Dehn's
results to higher dimensions. In this talk we will discuss this
construction and its connections to both algebraic and Hermitian
K-theory, and discuss the new avenues of attack that this presents for
the generalized Hilbert's third problem.